- Email: [email protected]
Photoemission calculations from ferromagnetic Ni(111) at elevated temperatures
Physica B 161 (1989) 169-171 North-Holland, Amsterdam
PHOTOEMISSION TEMPERATURES
H. GOLLISCH Theoretische
CALCULATIONS
FROM FERROMAGNETIC
Ni( 111)
AT ELEVATED
and R. FEDER
Festkiirperphysik,
Universitiit Duisburg-GH,
D-4100 Duisburg,
FRG
A correlated-local-moment theory of ferromagnetism at finite temperatures is applied to calculate off-normal photoemission spectra from Ni(1 1 1). Comparison with experimental data suggests that around the Curie temperature there are sizeable local moments with substantial ferromagnetic order.
The current understanding of 3d-metal ferromagnetism at finite temperatures is highlighted by the existence of local magnetic moments and a short-range order between them even above the Curie temperature (cf. [l-3], and references therein). The local moment vector m, associated with lattice site Ri is defined via pure quantum state expectation values of the spin density operator integrated over the lattice cell around R, (for details cf., e.g., chapter 1 in ref. [3]). Short-range order is described ([4]) by the moment-moment correlation function
where (. . .) is the thermostatistical ensemble average. It is conveniently characterized by the full width A at half-maximum of T(R,, Ri + R), i.e., for R = Al2 one has r = l/2. A powerful approach to determining the site-averaged size k = C 1mi ( /N of the moments and the amount of order A consists in comparing experimental angle-resolved photoemission spectra with their theoretical counterparts calculated by a correlated-local-moment theory, in which p and A are taken as adjustable input parameters. This approach has been applied to Ni [4], Fe [5-71 and the invar alloy Fe,Pt [8]. In the present paper, we extend the analysis of the Ni(1 1 1) photoemission data [9] given by Haines et al. [4] in two respects. Firstly, we calculate for the off-symmetry-line k point corresponding more closely to the experimental situation, and secondly we also include spectra below
T,. After some brief remarks on the theory, we present and discuss numerical results. To describe the electronic and magnetic structure at elevated temperatures, we employ a correlated-local-moment theory, in which spinresolved densities of states and photoemission intensities are obtained as averages of the respective quantities calculated for frozen cluster configurations of effective local moments of varying size and direction. Since this theory has been presented earlier [4-81, it may suffice to point out some modifications and assumptions specific for the present study. In the Pauli-like tight binding Hamiltonian, we allow the effective exchange field to be different for d states of eR and t,, symmetry. This introduces a k dependence of the exchange splitting. Parameter values chosen to match experimental band structures at lower temperature are A = 0 for s, p, and A = 0.165 and 0.33 eV for d and d orbitals. For the inverse-hole lifetim: we aszmed the form y = pE2/(E2 + EE) with p = 2 eV and E, = 2 eV. With regard to the cluster geometry, we departed from the traditional cubic form and rather chose a finite stack of (1 1 1) monatomic layers in order to facilitate the use of periodic boundary conditions. A total of 1400 atoms was used. The experimental photoemission spectra from ref. [9] -obtained for hw = 7.7 eV and polar emission angle 19= 45” - can for low temperatures be interpreted in terms of direct transitions originating from exchange-split bulk initial states with k around (1.075, 1.075, 0.43) rr/u. Calcula-
0921-4526/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
80 A cln
60
II 3
40
g
2o
cl V x
80
z!
60
+Q) s -
40
c
I
,.‘.
-1
20
I 80
_
T/‘Tc=O.60
80
_
T/Tc=O.48
-0.9
-0.7
-0.5
-0.3
Energy
-0.1
0.1
(eV)
Fig. 1. Photoemission from Ni(t 1 1) for hw = 7.7 eV and 19, = 4.5” (corresponding to k = (1.075, 1.075, 0.43) n/a) for temperatures and relative magnetization M as indicated. Experimental spectra from ref. [91 (xxx) and theoretical spin-summed ones for short-range-order parameter .,I = 0 (. .), 5.86 8, (--) and 10.15 A(- - -). For T < 7‘, thin solid lines give the theoretical spin-resolved spectra
, the for
n = 5.86A.
tions were performed accordingly. Since at this low photon energy the final-state mean free path is large, we chose for the average magnetization at a given temperature the bulk value. To take account of experimental energy and angular resolution and, to some extent, of lattic vibration effects, spectra were convoluted by a Gaussian of 0.15 eV FWHM. Figure 1 shows theoretical spectra for A = 0, 5.86A and 10.15~% corresponding to no, moderate and large short-range order,
respectively. The local moment was assumed to have the same magnitude for all sites and tempcratures. At room temperature (T= 0.48 T,.). long-range order dominates, leading to a majority and a minority spin peak with an exchange splitting of 0.28eV. With increasing 7’, the two peaks move towards each other and merge at T,. into a single peak, which is significantly broader than the spin-resolved peaks at room temperature. (Note that a temperature-dependence of lattice-vibrational broadening has not been taken into account). The broadening is due to the existence of local moments p near T,.. In the Stoner theory, where p = 0 for T 2 T,., the width of the peak would be close to the average width of the low-7‘ peaks. The merging of the two peaks with increasing T is found for .I = 0 and 5.86 A. while for 1I = 10.15 A the T = 1.01 T,. spectrum has a shoulder near 1.3 eV indicating exchange-split peaks. Comparison with the experimental spectra from [9] shows rather good agreement. bearing in mind that the calculated spectra do not include the inelastic current, which increases with decreasing E. The calculated minimun for low 7 at 2.7 eV rises closer to experiment if one assumes a stronger inverse hole-lifetime. From the two high-temperature panels it appears that *1 should be between about 5 and 9 A. This range agrees well with the estimate obtained by Haines ct al. [4]. Compared to Fe [S, 61. the present Ni results are relatively insensitive to variations of ‘4. in line with the trend towards merging-peak behaviour with decreasing (low-r) exchange splitting reported in ref. (41. To determine A more accurately. we suggest to explore photoemission or its inverse at a number of further k points, in the hope of finding different temperature behaviours. Acknowledgements This work was funded by the Deutsche Forschungsgemeinschaft in its Sonderforschungsbereich 166. Also, we gratefully acknowledge the kind hospitality of the Institut fiir Festkoperforschung of the KFA Jiilich.
H. Gollisch and R. Feder
I Photoemission calculations from ferromagnetic Ni(l 1 1) [.51E.M.
References Capellmann, chapter 2 in ref. [2]. Electrons in Surface Physics, M Polarized (World Scientific, Singapore, 1985). H. Capellmann, [31Metallic Magnetism, Heidelberg, 1987). 141E.M. Haines, V. Heine and A. Ziegler, (1986) 1343.
[6]
111H.
R. Feder, ed.
ed.
(Springer,
J. Phys.
F 16
[7] [S] [9]
171
Haines, R. Clauberg and R. Feder. Phvs. Rev. Lett. 54 (1985) 932. R. Clauberg, E.M. Haines and R. Feder, Z. Phys. B 62 (1985) 31. E.M. Haines, R. Clauberg, E. Tamura and R. Feder, Solid State Commun. 57 (1986) 669. H. Gollisch and R. Feder, Solid State Commun. 69 (1989) 579. C.J. Maetz, U. Gerhardt, E. Dietz, A. Ziegler and R.J. Jelitto, Phys. Rev. Lett. 48 (1982) 1686.
PHOTOEMISSION TEMPERATURES
H. GOLLISCH Theoretische
CALCULATIONS
FROM FERROMAGNETIC
Ni( 111)
AT ELEVATED
and R. FEDER
Festkiirperphysik,
Universitiit Duisburg-GH,
D-4100 Duisburg,
FRG
A correlated-local-moment theory of ferromagnetism at finite temperatures is applied to calculate off-normal photoemission spectra from Ni(1 1 1). Comparison with experimental data suggests that around the Curie temperature there are sizeable local moments with substantial ferromagnetic order.
The current understanding of 3d-metal ferromagnetism at finite temperatures is highlighted by the existence of local magnetic moments and a short-range order between them even above the Curie temperature (cf. [l-3], and references therein). The local moment vector m, associated with lattice site Ri is defined via pure quantum state expectation values of the spin density operator integrated over the lattice cell around R, (for details cf., e.g., chapter 1 in ref. [3]). Short-range order is described ([4]) by the moment-moment correlation function
where (. . .) is the thermostatistical ensemble average. It is conveniently characterized by the full width A at half-maximum of T(R,, Ri + R), i.e., for R = Al2 one has r = l/2. A powerful approach to determining the site-averaged size k = C 1mi ( /N of the moments and the amount of order A consists in comparing experimental angle-resolved photoemission spectra with their theoretical counterparts calculated by a correlated-local-moment theory, in which p and A are taken as adjustable input parameters. This approach has been applied to Ni [4], Fe [5-71 and the invar alloy Fe,Pt [8]. In the present paper, we extend the analysis of the Ni(1 1 1) photoemission data [9] given by Haines et al. [4] in two respects. Firstly, we calculate for the off-symmetry-line k point corresponding more closely to the experimental situation, and secondly we also include spectra below
T,. After some brief remarks on the theory, we present and discuss numerical results. To describe the electronic and magnetic structure at elevated temperatures, we employ a correlated-local-moment theory, in which spinresolved densities of states and photoemission intensities are obtained as averages of the respective quantities calculated for frozen cluster configurations of effective local moments of varying size and direction. Since this theory has been presented earlier [4-81, it may suffice to point out some modifications and assumptions specific for the present study. In the Pauli-like tight binding Hamiltonian, we allow the effective exchange field to be different for d states of eR and t,, symmetry. This introduces a k dependence of the exchange splitting. Parameter values chosen to match experimental band structures at lower temperature are A = 0 for s, p, and A = 0.165 and 0.33 eV for d and d orbitals. For the inverse-hole lifetim: we aszmed the form y = pE2/(E2 + EE) with p = 2 eV and E, = 2 eV. With regard to the cluster geometry, we departed from the traditional cubic form and rather chose a finite stack of (1 1 1) monatomic layers in order to facilitate the use of periodic boundary conditions. A total of 1400 atoms was used. The experimental photoemission spectra from ref. [9] -obtained for hw = 7.7 eV and polar emission angle 19= 45” - can for low temperatures be interpreted in terms of direct transitions originating from exchange-split bulk initial states with k around (1.075, 1.075, 0.43) rr/u. Calcula-
0921-4526/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
80 A cln
60
II 3
40
g
2o
cl V x
80
z!
60
+Q) s -
40
c
I
,.‘.
-1
20
I 80
_
T/‘Tc=O.60
80
_
T/Tc=O.48
-0.9
-0.7
-0.5
-0.3
Energy
-0.1
0.1
(eV)
Fig. 1. Photoemission from Ni(t 1 1) for hw = 7.7 eV and 19, = 4.5” (corresponding to k = (1.075, 1.075, 0.43) n/a) for temperatures and relative magnetization M as indicated. Experimental spectra from ref. [91 (xxx) and theoretical spin-summed ones for short-range-order parameter .,I = 0 (. .), 5.86 8, (--) and 10.15 A(- - -). For T < 7‘, thin solid lines give the theoretical spin-resolved spectra
, the for
n = 5.86A.
tions were performed accordingly. Since at this low photon energy the final-state mean free path is large, we chose for the average magnetization at a given temperature the bulk value. To take account of experimental energy and angular resolution and, to some extent, of lattic vibration effects, spectra were convoluted by a Gaussian of 0.15 eV FWHM. Figure 1 shows theoretical spectra for A = 0, 5.86A and 10.15~% corresponding to no, moderate and large short-range order,
respectively. The local moment was assumed to have the same magnitude for all sites and tempcratures. At room temperature (T= 0.48 T,.). long-range order dominates, leading to a majority and a minority spin peak with an exchange splitting of 0.28eV. With increasing 7’, the two peaks move towards each other and merge at T,. into a single peak, which is significantly broader than the spin-resolved peaks at room temperature. (Note that a temperature-dependence of lattice-vibrational broadening has not been taken into account). The broadening is due to the existence of local moments p near T,.. In the Stoner theory, where p = 0 for T 2 T,., the width of the peak would be close to the average width of the low-7‘ peaks. The merging of the two peaks with increasing T is found for .I = 0 and 5.86 A. while for 1I = 10.15 A the T = 1.01 T,. spectrum has a shoulder near 1.3 eV indicating exchange-split peaks. Comparison with the experimental spectra from [9] shows rather good agreement. bearing in mind that the calculated spectra do not include the inelastic current, which increases with decreasing E. The calculated minimun for low 7 at 2.7 eV rises closer to experiment if one assumes a stronger inverse hole-lifetime. From the two high-temperature panels it appears that *1 should be between about 5 and 9 A. This range agrees well with the estimate obtained by Haines ct al. [4]. Compared to Fe [S, 61. the present Ni results are relatively insensitive to variations of ‘4. in line with the trend towards merging-peak behaviour with decreasing (low-r) exchange splitting reported in ref. (41. To determine A more accurately. we suggest to explore photoemission or its inverse at a number of further k points, in the hope of finding different temperature behaviours. Acknowledgements This work was funded by the Deutsche Forschungsgemeinschaft in its Sonderforschungsbereich 166. Also, we gratefully acknowledge the kind hospitality of the Institut fiir Festkoperforschung of the KFA Jiilich.
H. Gollisch and R. Feder
I Photoemission calculations from ferromagnetic Ni(l 1 1) [.51E.M.
References Capellmann, chapter 2 in ref. [2]. Electrons in Surface Physics, M Polarized (World Scientific, Singapore, 1985). H. Capellmann, [31Metallic Magnetism, Heidelberg, 1987). 141E.M. Haines, V. Heine and A. Ziegler, (1986) 1343.
[6]
111H.
R. Feder, ed.
ed.
(Springer,
J. Phys.
F 16
[7] [S] [9]
171
Haines, R. Clauberg and R. Feder. Phvs. Rev. Lett. 54 (1985) 932. R. Clauberg, E.M. Haines and R. Feder, Z. Phys. B 62 (1985) 31. E.M. Haines, R. Clauberg, E. Tamura and R. Feder, Solid State Commun. 57 (1986) 669. H. Gollisch and R. Feder, Solid State Commun. 69 (1989) 579. C.J. Maetz, U. Gerhardt, E. Dietz, A. Ziegler and R.J. Jelitto, Phys. Rev. Lett. 48 (1982) 1686.